Summary: A Purely Combinatorial Proof of the Hadwiger Debrunner
(p, q) Conjecture
N. Alon , Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact
Sciences, Tel Aviv University, Tel Aviv, Israel and Institute for Advanced Study,
Princeton, NJ 08540, USA. Email: noga@math.tau.ac.il.
D.J. Kleitman ,Department of Mathematics, MIT, Cambridge, MA, 02139. Email:
djk@math.mit.edu.
Submitted: July, 1996; Accepted: December, 1996.
Abstract
A family of sets has the (p, q) property if among any p members of the family some
q have a nonempty intersection. The authors have proved that for every p q d + 1
there is a c = c(p, q, d) < such that for every family F of compact, convex sets in
Rd
which has the (p, q) property there is a set of at most c points in Rd
that intersects
each member of F, thus settling an old problem of Hadwiger and Debrunner. Here we
present a purely combinatorial proof of this result.
AMS Subject Classification: 52A35
1. Introduction
The purpose of this note is to present an elementary and self contained description of